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ALMOST
-NORMAL AND MILDLY
-NORMAL
SPACES IN TOPOLOGICAL SPACES
Hamant Kumar and M. C. Sharma
Department of Mathematics, N. R. E. C. College, Khurja-203131
ABSTRACT
The aim of this paper is to introduce and study two new classes of spaces, namely almost -normal and mildly -normal spaces are weaker forms of --normal spaces. We show that these -normal spaces, namely almost --normal and mildly -normal spaces are regularly open hereditary. The relationships among normal, p-normal, -normal, -normal, almost normal, almost p-normal, almost -normal, almost -normal, quasi -normal, mildly normal,
mildly p-normal, mildly -normal, and mildly -normal spaces are investigated. Moreover, we introduced some functions such as M--open, M--closed, almost -irresolute, rg-irresolute, rg-continuous and -rg-continuous. Further, utilizing gclosed and rgclosed sets, we obtain characterizations and preservation theorems for almost -normal and mildly --normal spaces.
Key words:
-open sets, almost
-normal, mildly
-normal spaces, M-
-open, M-
-closed, almost
-irresolute, rg
-irresolute, rg
-continuous and
-rg
-continuous functions.
2010 Mathematics subject classification: 54D10, 54D15, 54A05, 54C08.
I INTRODUCTION
Levine [6] initiated the study of so called generalized closed (briefly g-closed) sets in order to extend many of the most important properties of closed sets to a large family. Singal and Arya [18] introduced the concept of almost normal spaces. Various properties of new classes of topological spaces have been studied and the relations of these new concepts with the concepts of almost regularity have also been investigated. The notion of mildly normal space was introduced by Shchepin [17] and Singal and Singal [19] independently. Nour [13] introduced a weaker form of normality, called p-normality and obtained their properties. Mahmoud et al. [7] introduced the notion of -normal
spaces and obtained their characterizations and preservation theorems. E. Ekici [3] introduced a weaker form of normality, called -normality and obtained their properties. The relationships among normal, p-normal, s-normal and -normal spaces are investigated. The notion of quasi -normal and mildly –normal spaces were introduced by M.
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II PRELIMINARIES
Throughout this paper, spaces (X, ), (Y, ), and (Z, ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a space X. The closure of A and interior of A are
denoted by cl(A) and int(A) respectively. A subset A is said to be regular open (resp. regular closed) if A int(cl(A)) (resp. A cl(int(A)). A is said to be -open [1] if A cl(int(cl(A))), preopen [8] (briefly p-open) if
A int(cl(A)), -open [4] if A cl(int(A)) int(cl(A)). The complement of a –open (resp. p-open) set is said to be –closed [1] (resp. p-closed [8]). The intersection of all –closed (resp. p-closed, -closed) sets containing A is
called –closure [1] (resp. p-closure, -closure) of A, and is denoted by cl(A) (resp. pcl(A), cl(A)). The -Interior of A, denoted by int(A), is defined as union of all -open sets contained in A.
2.1. Definition.
A subset A of a topological space X is said to be1. generalized closed [6] (briefly g-closed) if cl(A) U whenever A U and U is open in X.
2. regularly generalized [14] (briefly rg-closed) if cl(A) U whenever A U and U is regularlyopen in X. 3. generalized pre-closed [11] (briefly gp-closed) if pcl(A) U whenever A U and U is open in X.
4. generalized preregularly-closed [5] (briefly gpr-closed) if pcl(A) U whenever A U and U is regularlyopen in X.
5. generalized -closed [3] (briefly g-closed) if cl(A) U whenever A U and U is open in X.
6. regularly generalized -closed (briefly rg-closed) if cl(A) U whenever A U and U is regularlyopen in X. 7. generalized -closed [2] (briefly g-closed) if cl(A) U whenever A U and U is open in X.
6. regularly generalized -closed [15] (briefly rg-closed) if cl(A) U whenever A U and U is regularlyopen in X.
The complement of g-closed, (resp. rg-closed, gp-closed, gpr-closed, g-closed, rg-closed, g-closed, rg-closed)
sets is g-open (resp. rg-open, gp-open, gpr-open, g-open, rg-open, g-open, rg-open) sets.
By the definitions stated above and in preliminaries. We have the following diagram:
closed
g-closed
rg-closed
p-closed
gp-closed
gpr-closed
-closed
g
-closed
rg
-closed
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Where none of the implications is reversible as can be seen from the following examples:
2.2. Example. Let X = {a, b, c, d} and = { , {a}, {b, c}, {a, b, c}, X}. Then A = { a } is g-closed but not
closed.
2.3. Example. Let X = {a, b, c, d} and = {, {b, d}, {a, b, d}, {b, c, d}, X}. Then A = {a, b} is g-closed as well as
rg-closed but not closed.
2.4. Example. Let X = {a, b, c, d} and = {, {b}, {d}, {b, d}, X}. Then A = {a, b, d} is g-closed but not closed.
2.5. Example. Let X = {a, b, c} and = {, {b}, {c}, {b, c}, X}. Then A = {b, c} is gpr-closed as well as rg-closed, but not p-closed.
2.6. Example. Let X = {a, b, c, d, e} and = {, {a, b}, {c, d}, {a, b, c, d}, X}. Then A = {a} gpr-closed as well as
rg-closed, but not rg-closed.
2.7. Example. Let X = {a, b, c} and = {, {a}, {b},{a, b}, X}. Then A = {a} is g-closed. But it is neither closed
nor gp-closed.
2.8. Example. Let X = {a, b, c} and = {, {b, c}, X}. Then A = {c} is pre-closed but not closed. 2.9. Example. Let X = {a, b, c} and = {, {a}, X}. Then A = {b} is g-closed but not closed.
2.10. Example. Let X = {a, b, c,} and = {, {a}, {a, b}, X}. Then A = {b} is g-closed but not g-closed.
III ALMOST
-NORMAL SPACES
3.1. Definition. A topological space X is said to be almost -normal (resp. almost normal [18], almost p-normal
[9], almost -normal [10]) if for any two disjoint closed subsets A and B of X, one of which is regularly closed, there exist disjoint -open (resp. open, p-open, -open) sets U and V of X such that A U and B V.
3.2. Definition. A topological space X is said to be -normal [3] (resp. p-normal [13], -normal [7]) if for every pair of disjoint closed subsets A, B of X, there exist disjoint -open (resp. p-open, -open) sets U, V of X such that A U and B V.
By the definitions stated above and in preliminaries. We have the following diagram:
normal
almost normal
p-normality
almost p-normality
-normality
almost
-normality
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Where none of the implications is reversible as can be seen from the following examples:
3.3 Example. Let X = {a, b, c, d} and = {, {a}, {b, d}, {a, b, d}, {b, c, d}, X}. The pair of disjoint closed subsets
of X are A = {a} and B = {c}. Also U = {a, b} and V = {c, d} are p-open sets such that A U and B V. Hence the space X is p-normal as well as -normal but not normal, since U and V are not open sets.
3.4 Example. Let X = {a, b, c, d} and = {, {b, d}, {a, b, d}, {b, c, d}, X}. The pair of disjoint closed subsets of
X are A = {a} and B = {c}. Also U = {a, b} and V = {c, d} are -open sets such that A U and B V. Hence the space X is -normal but not normal, since U and V are not open sets.
3.5 Example. Let X = {a, b, c, d} and = {, {b}, {d}, {b, d}, {a, b, d}, {b, c, d}, X}. The pair of disjoint closed subsets of X are A = {a} and B = {c}. Also U = {a, b} and V = {c, d} are -open sets such that A U and B V. Hence the space X is -normal. But it is neither p-normal nor normal, since U and V are neither p-open nor open
sets.
3.6 Example. Let X = {a, b, c, d, e} and = {, {a}, {c}, {a, c}, {b, c}, {c, e}, {a, b, c}, {a, c, e}, {b, c, d}, {b, c, e}, {a, b, c, d}, {a, b, c, e}, {b, c, d, e}, X}. The pair of disjoint closed subsets of X are A = {a} and B = {d}. Also U = {a} and V = {b, c, d} are open sets such that A U and B V. Hence the space X is normal as well as -normal, since U and V are also -open sets.
3.7 Example. Let X = {a, b, c,} and = {, {a}, {b, c}, X}. The pair of disjoint closed subsets of X are A = {a} and
B = {b, c}. Also U = {a} and V = {b, c} are open sets such that A U and B V. Hence the space X is normal as
well as almost normal.
3.8 Example. Let X = {a, b, c, d} and = {, {a}, {b}, {d}, {a, b}, {a, d}, {b, d}, {a, b, c}, {a, b, d}, X}. The pair
of disjoint closed subsets of X are A = {a, b, c} and B = {d}. Also U = {a, b, c} and V = {d} are open sets such that A U and B V. Hence the space X is normal as well as -normal.
3.9. Theorem. A X is rg-open if and only if F int (A) whenever F is regularly closed and F
Proof. Let A be rg-open. Let F be regularly closed and F A. Then X – A X – F. X – F is regularly open.
X – A is rg-closed. cl(X – A) X – F. X – int(A) X – F. So, F int(A). Let F regularly closed and F A imply F int(A). Let X – A U, where U is regularly open X – U A, where X – U is regularly closed. Hence
X – U int(A). So X – int(A) U. That is cl(X – A) U. Hence X – A is rg-closed. This implies A is rg-open.
3.10. Theorem. For a topological space X, the following are equivalent: (a) X is almost -normal.
(b) For every closed set A and every regularly open set B containing A, there is a -open set U such that
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(c) For every regularly closed set A and every open set B containing A, there is a -open set U such that
A U cl(U) B.
(d) For every pair of disjoint sets A and B of X, one of which is closed and other is regularly closed, there exist
-open sets U and V such that A U, B V and cl(U) cl(V) = .
Proof. (a) (b). Let A be a closed set and let B be a regularly open set containing A. Thus A (X – B) = , where
A is closed and X – B is regularly closed. Therefore, there exist -open sets U and V such that A U, X – B V and U V = . Thus, A U X – V B. Now, X – V is closed. Therefore, A U cl(U) B.
(b) (c).Let A be regularly closed set and let B be an open set containing A. Then, X – B X – A, whence X – A
is a regularly open set containing the closed set X – B. Therefore, there is a -open set M such that X – B M cl(M) X – A. Thus, A X – cl(M) X – M B. Let X – cl(M) = U. Then U is -open and A U cl(U) B.
(c) (d). Let A be a regularly closed set and B be a closed set such that A B = . Then, A X – B which is
open. Therefore, there exists a -open set M such that A M cl(M) X – B. Again, M is a -open set containing
the regularly closed set A. Therefore, there is a -open set U such that A U cl(U) M. Let X – cl(M) = V. Then, A U, B V and cl(U) cl(V) = .
(d) (a) is obvious.
3.11 Theorem. For a topological space X, the following are equivalent: (a) X is almost -normal.
(b) For every closed set A and every regularly closed set B, there exist disjoint g-open sets U and V such that A U and B V.
(c) For every closed set A and every regularly closed set B, there exist disjoint rg-open sets U and V such that A U and B V.
(d) For every closed set A and every regularly open set B containing A, there exists a g-open set U of X such that A U -cl(U) B.
(e) For every closed set A and every regularly open set B containing A, there exists a rg-open set U of X such that A U -cl(U) B.
(f) For every pair of disjoint sets A and B of X , one of which is closed and other is regularly closed, there exist -open sets U and V such that A U and B V and U V =
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(a) (b). Let X be almost -normal. Let A be a closed and B be a regularly closed sets in X. By assumption, there
exist disjoint -open sets U and V such that A U and B V. Since every -open set is g-open set, U, V are g-open sets such that A U and B V.
(b) (c). Let A be a closed and B be a regularly closed sets in X. By assumption, there exist disjoint g-open sets U and V such that A U and B V. Since every g-open set is rg-open set, U, V are rg-open sets such that A U and B V.
(d) (e). Let A be any closed set and B be any regularly open set containing A. By assumption, there exists a g-open set U of X such that A U cl(U) B. Since every g-open set is rg-open set, there exists a rg-open set U of X such that A U cl(U) B.
(c) (d). Let A be any closed set and B be a regularly open set containing A. By assumption, there exist disjoint rg-open sets U and W such that A U and X – B W. By Theorem 3.9, we get, X – B int(W) and cl(U) int(W) = . Hence A U cl(U) X– int(W) B.
(e) (f). For any closed set A and any regularly open set B containing A. Then A X – B and X – B is a regularly closed set. By assumption, there exists a rg-open set G of X such that A G cl(G) X – B. Put U = int(G), V = X – cl(G). Then U and V are disjoint -open sets of X such that A U and B V.
(f) (a) is obvious.
IV PRESERVATION THEOREMS
4.1 Definition. A function f : X Y is called
1. M--open if f(U) O(Y) for each U O(X) .
2. M--closed if f(U) C(Y) for each U C(X) .
3. almost -irresolute if for each x X and each -neighbourhood V of f(x), -cl (f –1(V) ) is a -neighbourhood
of x.
4.2 Definition. A topological space X is called
1. weakly -regular if for each point x and a regularly open set U containing {x}, there is a -open set V such that
x V -cl(V) V.
2. almost -regular if for every regularly closed set F and each point x F, there exist disjoint -open sets U and V
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4.3 Remark. One can prove that almost -normality is also regularly closed hereditary. 4.4 Remark. Almost -normality does not imply almost -regular.
Next, we prove the invariant of almost -normality in the following.
4.5 Theorem. If f : X Y is continuous M--open rc-continuous and almost -irresolute surjection from an almost
-normal space X onto a space Y, then Y is almost -normal.
Proof. Let A be a closed set and B be a regularly open set containing A. Then by rc-continuity of f, f –1(A) is a closed set contained in the regularly open set f –1(B). Since X is almost -normal, there exist a -open set V in X such that f –1(A) V -cl(V) f –1(B) by Theorem 3.11. Then, f(f –1(A)) f(V) f(-cl(V)) f(f –1(B)). Since f is M--open and almost -irresolute surjection, we obtain a A f(V) -cl(f(V)) B. Then again by Theorem 3.11, Y is almost -normal.
4.6 Theorem.If f : X Y is rc-continuous M- -closed map from an almost -normal space X onto a space Y, then Y is almost -normal.
Proof. Easy to verify.
V MILDLY
-NORMAL SPACES
5.1 Definition. A topological space X is said to be mildly -normal (resp. mildly normal [17, 19], mildly p-normal [9], mildly -normal [16]) if for any two disjoint regularly closed subsets A and B of X, there exist two disjoint -open (resp. open, p-open, -open) subsets U and V of X such that A U and B V.
By the definitions stated above and in preliminaries. We have the following diagram:
normal
almost normal
mildly normal
p-normal
almost p-normal
mildly p-normal
-normal
almost
-normal
mildly
-normal
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Where none of the implications is reversible as can be seen from the following examples:
5.2 Example. Let X = {a, b, c, d, e} and = {, {a}, {c, d}, {a, c, d}, {b, c, d, e}, X}. The pair of disjoint closed
subsets of X are A = {a} and B = {b, e}. Also U = {a} and V = {b, c, d, e} are open sets such that A U and B V. Hence the space X is normal as well as -normal.
5.3. Example. Let X = {a, b, c} and = {, {a, b}, {c}, X}. The pair of disjoint regularly closed subsets of X are
A = {a, b} and B = {c} . Also U = {a, b} and V = {c} are open sets such that A U and B V. Hence the space X is mildly normal as well as mildly -normal, since U and V are also -open sets.
5.4 Example. Let X = {a, b, c, d} and = {, {c}, {d}, {a, b}, {c, d}, {a, b, d}, {a, b, c}, X}. The pair of disjoint regularly closed subsets of X are A = {c} and B = {d}. Also U = {b, c} and V = {a, d} are -open sets such that A U and B V. Hence the space X is mildly -normal. But the space X is neither mildly normal nor normal, since U and V are not open sets.
5.5 Theorem. For a topological space X the following are equivalent: (a) X is mildly -normal.
(b) For every pair of regularly open sets U and V of X whose union is X, there exist -closed sets A and B such that A U, B V and AB =X
(c) For every regularly closed set F and every regularly open set G containing F, there exists a -open set U such that F U cl(U) G.
Proof. (a) (b).Let U and V be a pair of regularly open sets in a mildly -normal space X such that X = U V.
Then X – U, X – V are disjoint regularly closed sets. Since X is mildly -normal, there exist disjoint -open sets U1
and V1 such that X – U U1 and X – V V1. Let A = X – U1, B = X – V1. Then A and B are -closed sets such that A U, B V and A B = X.
(b) (c). Let F be a regularly closed set and G be a regularly open set containing F. Then X – F and G are regularly open sets whose union is X. Then by (b), there exist -closed sets W1 and W2 such that W1 X – F and W2 G and
W1 W2 = X. Then F X – W1, X – G X – W2 and (X – W1) (X – W2) = . Let U = X – W1 and V = X – W2. Then U and V are disjoint -open sets such that F U X – V G. As X – V is -closed set, we have cl(U) X – V and F U cl(U) G.
(c) (a). Let F1 and F2 be any two disjoint regularly closed sets of X. Put G = X – F2, then F1 G = . F1 G,
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follows that F2 X – cl(U) = V, says, then V is -open and U V = . Hence F1 and F2 are separated by -open sets U and V. Therefore X is mildly -normal.
5.6 Theorem. For a topological space X, the following are equivalent: (a) X is mildly -normal.
(b) For any disjoint regularly closed sets A and B of X, there exist disjoint g-open sets U and V such that A U and B V.
(c) For any disjoint regularly closed sets A and B of X, there exist disjoint rg-open sets U and V such that A U and B V.
(d) For any regularly closed set A and any regularly open set V containing A, there exists a g-open set U of X such that H U cl(U) V.
(e) For any regularly closed set A and any regularly open set V containing A, there exists a rg-open set U of X such that A U cl(U) V.
Proof. (a) (b), (b) (c), (c) (d), (d) (e) and (e) (a).
(a) (b). Let X be mildly -normal space. Let A, B be disjoint regularly closed sets of X. By assumption, there
exist disjoint -open sets U, V such that A U and B V. Since every -open set is g-open, so, U and V are g-open sets such that H U and K V.
(b) (c). Let A, B be two disjoint regularly closed sets. By assumption, there exist g-open sets U and V such that A U and B V. Since every g-open set is rg-open, so, U and V are rg-open such that H U and K V.
(c) (d). Let A be any regularly closed set and V be any regularly open set containing A. By assumption, there exist rg-open sets U and W such that A U and X – V W. By Theorem 3.9, we get X – V int(W) and
U int (W) = Therefore, we obtaincl(U) int(W) = and hence A U cl(U) X int(W) V.
(d) (e). Let A be any regularly closed set and V be any regularly open set containing A. By assumption, there exist g-open set U of X such that H U cl(U) V. Since, every g-open set is rg-open, there exist rg-open sets U of X such that H U cl(U) V.
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Using Theorem 5.6, it is easy to show the following theorem, which is a Urysohn’s Lemma version for mild -normality. A proof can be established by a similar way of the normal case.
5.7 Theorem. A space X is mildly -normal if and only if for every pair of disjoint regularly closed sets A and B of
X, there exists a continuous function f on X into [0, 1], with its usual topology, such that f(A) = {0} and f(B) = {1}.
5.8 Theorem. Let X is a mildly -normal space and f : X Y is an open continuous injective function. Then f(X) is
a mildly -normal space.
Proof. Let A and B be any two regularly closed subset of f(X) such that A ∩ B = . Then f –1(A) and f –1(B)
regularly closed sets of X. Since X is mildly -normal, there are two disjoint -open sets U and V such that f –1(A) U and f –1(B) V. Since f is one-one and open, result follows.
5.9 Corollary. Mild -normality is a topological property.
5.10 Definition. A function f : X Y is said to be
1. rg-continuous if f –1(F) is rg-closed in X for every closed set F of Y. 2. -rg-continuous if f –1(F) is rg-closed in X for every -closed set F of Y. 3. rg-irresolute if f –1(F) is rg-closed in X for every rg-closed set F of Y.
5.11 Theorem. If f : X Y is a -rg-continuous, rc-preserving injection and Y is mildly -normal then X is mildly
-normal.
Proof. Let A and B be any disjoint regularly closed sets of X. Since f is an rc-preserving injection, f(A) and f(B) are disjoint regularly closed sets of Y. By mild -normality of Y, there exist disjoint -open sets U and V of Y such that f(A) U and f(B) V. Since f is -rg-continuous, f –1(U) and f –1(V) are disjoint rg-open sets containing A and
B respectively. Hence by Theorem 5.6, X is mildly -normal.
5.12 Theorem. If f : X Y is a -rg-continuous, almost closed surjection and Y is -normal space, then X is mildly -normal.
Proof. Similar to previous one.
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